Wringer roller mops are well-known in the industry. Such devices include a cleaning head in the form of a polyurethane or polyester sponge. To wring the sponge, the sponge is drawn between a pair of rollers that straddle the sponge. Although the rollers are effective in wringing most of the liquid from the sponge, a certain amount of liquid remains in the sponge after wringing so that the sponge may tend to push liquid around on a surface rather than absorbing the liquid. Accordingly, when using a wringer roller mop, frequently an undesirable amount of residual fluid is left on the surface being cleaned.